Loss Data Analysis

9Superresolution in maxentropic Laplace transform inversion

9.1Introductory remarks

We have proposed the method of maximum entropy to determine a probability density of a positive random variable from the knowledge of its Laplace transform. In Chapter 4 we provided several examples of situations in which the problem appears, and in Chapters 7 and 8 we presented a collection of numerical examples in which we applied the method. The problem was stated as

$\text{Find}{f}_S\left(s\right)\text{such}\text{that}\text{E}\left[e^{-S{\alpha }_i}\right]={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-s{\alpha }_i}{f}_S\left(s\right)ds={\mu }_i;}i=1,\dots ,K.\left(9.1\right)$

In Chapter 6 we mentioned that this problem can be solved by transforming it into a fractional moment problem on [0, 1], which can then be solved applying the maximum entropy technique. In the examples considered ...